Place 1 tablespoon of butter in a skillet over medium heat. Once melted, add the sliced bananas and sprinkle with 1/4 cup brown sugar. Cook for about 4-5 minutes, gently stirring occasionally, until the bananas are golden and caramelized. Remove from heat and set aside to cool slightly.
Preheat your oven to 325°F (163°C). In a small bowl, mix the graham cracker crumbs with the melted 1/4 cup unsalted butter until the crumbs are evenly coated and resemble wet sand. Press this mixture firmly into the bottom of your 8x8-inch baking pan to form an even crust layer.
In a large mixing bowl, beat the softened cream cheese until smooth and creamy using an electric mixer or whisk. Gradually add the granulated sugar and beat until combined. Add the vanilla extract and eggs, one at a time, mixing well after each addition. Finally, stir in the flour to help the filling set properly.
Gently fold the caramelized bananas into the cheesecake batter, being careful not to break the banana slices too much. This will distribute the sweet, caramel flavor throughout the filling.
Pour the cheesecake mixture over the prepared crust in the baking pan. Smooth the top with a spatula. Bake in the preheated oven for 40-45 minutes, or until the edges are set but the center is still slightly jiggly. This ensures a creamy texture once cooled.
Allow the bars to cool on a cooling rack for about 1 hour. Then, place the pan in the refrigerator for at least 4 hours or overnight. Chilling is essential to let the cheesecake bars fully firm up, making them easier to slice.
Once chilled, cut into squares and serve. These bars shine on their own but also pair beautifully with a dollop of whipped cream or a drizzle of caramel sauce for extra indulgence.